Making a change of variables $u=\tan(\theta)$:
$$
\int_0^{\pi/2} \sin\left(\tan \theta\right) \mathrm{d} \theta = \int_0^\infty \frac{\sin(u)}{1+u^2} \mathrm{d} u \tag{1}
$$
In order to evaluate this we use the technique of Mellin transform.